Theorem unirnmapsn 39406

Description: Equality theorem for a subset of a set exponentiation, where the exponent is a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
unirnmapsn.A	⊢ (𝜑 → 𝐴 ∈ 𝑉)
unirnmapsn.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
unirnmapsn.C	⊢ 𝐶 = {𝐴}
unirnmapsn.x	⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑𝑚 𝐶))

Assertion

Ref	Expression
unirnmapsn	⊢ (𝜑 → 𝑋 = (ran ∪ 𝑋 ↑𝑚 𝐶))

Proof of Theorem unirnmapsn

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unirnmapsn.C	. . . . 5 ⊢ 𝐶 = {𝐴}
2		snex 4908	. . . . 5 ⊢ {𝐴} ∈ V
3	1, 2	eqeltri 2697	. . . 4 ⊢ 𝐶 ∈ V
4	3	a1i 11	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
5		unirnmapsn.x	. . 3 ⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑𝑚 𝐶))
6	4, 5	unirnmap 39400	. 2 ⊢ (𝜑 → 𝑋 ⊆ (ran ∪ 𝑋 ↑𝑚 𝐶))
7		simpl 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → 𝜑)
8		equid 1939	. . . . . . . . 9 ⊢ 𝑔 = 𝑔
9		rnuni 5544	. . . . . . . . . 10 ⊢ ran ∪ 𝑋 = ∪ 𝑓 ∈ 𝑋 ran 𝑓
10	9	oveq1i 6660	. . . . . . . . 9 ⊢ (ran ∪ 𝑋 ↑𝑚 𝐶) = (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)
11	8, 10	eleq12i 2694	. . . . . . . 8 ⊢ (𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶) ↔ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶))
12	11	biimpi 206	. . . . . . 7 ⊢ (𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶) → 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶))
13	12	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶))
14		ovexd 6680	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ↑𝑚 𝐶) ∈ V)
15	14, 5	ssexd 4805	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ V)
16		rnexg 7098	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝑋 → ran 𝑓 ∈ V)
17	16	rgen 2922	. . . . . . . . . . . 12 ⊢ ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V
18	17	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
19		iunexg 7143	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V) → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
20	15, 18, 19	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
21	20, 4	elmapd 7871	. . . . . . . . 9 ⊢ (𝜑 → (𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶) ↔ 𝑔:𝐶⟶∪ 𝑓 ∈ 𝑋 ran 𝑓))
22	21	biimpa 501	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → 𝑔:𝐶⟶∪ 𝑓 ∈ 𝑋 ran 𝑓)
23		unirnmapsn.A	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
24		snidg 4206	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
25	23, 24	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
26	25, 1	syl6eleqr 2712	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
27	26	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → 𝐴 ∈ 𝐶)
28	22, 27	ffvelrnd 6360	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → (𝑔‘𝐴) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓)
29		eliun 4524	. . . . . . 7 ⊢ ((𝑔‘𝐴) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↔ ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
30	28, 29	sylib 208	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
31	7, 13, 30	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
32		elmapfn 7880	. . . . . . . 8 ⊢ (𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶) → 𝑔 Fn 𝐶)
33	32	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → 𝑔 Fn 𝐶)
34		simp3 1063	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) ∈ ran 𝑓)
35	23	3ad2ant1 1082	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝐴 ∈ 𝑉)
36	1	oveq2i 6661	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ↑𝑚 𝐶) = (𝐵 ↑𝑚 {𝐴})
37	5, 36	syl6sseq 3651	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑𝑚 {𝐴}))
38	37	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑋 ⊆ (𝐵 ↑𝑚 {𝐴}))
39		simpr 477	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ 𝑋)
40	38, 39	sseldd 3604	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (𝐵 ↑𝑚 {𝐴}))
41		unirnmapsn.b	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
42	41	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝐵 ∈ 𝑊)
43	2	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → {𝐴} ∈ V)
44	42, 43	elmapd 7871	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑓 ∈ (𝐵 ↑𝑚 {𝐴}) ↔ 𝑓:{𝐴}⟶𝐵))
45	40, 44	mpbid 222	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓:{𝐴}⟶𝐵)
46	45	3adant3 1081	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓:{𝐴}⟶𝐵)
47	35, 46	rnsnf 39370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → ran 𝑓 = {(𝑓‘𝐴)})
48	34, 47	eleqtrd 2703	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) ∈ {(𝑓‘𝐴)})
49		fvex 6201	. . . . . . . . . . . . 13 ⊢ (𝑔‘𝐴) ∈ V
50	49	elsn 4192	. . . . . . . . . . . 12 ⊢ ((𝑔‘𝐴) ∈ {(𝑓‘𝐴)} ↔ (𝑔‘𝐴) = (𝑓‘𝐴))
51	48, 50	sylib 208	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) = (𝑓‘𝐴))
52	51	3adant1r 1319	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) = (𝑓‘𝐴))
53	23	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 Fn 𝐶) → 𝐴 ∈ 𝑉)
54	53	3ad2ant1 1082	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝐴 ∈ 𝑉)
55		simp1r 1086	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 Fn 𝐶)
56	40, 36	syl6eleqr 2712	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (𝐵 ↑𝑚 𝐶))
57		elmapfn 7880	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (𝐵 ↑𝑚 𝐶) → 𝑓 Fn 𝐶)
58	56, 57	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐶)
59	58	adantlr 751	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐶)
60	59	3adant3 1081	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓 Fn 𝐶)
61	54, 1, 55, 60	fsneq 39398	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔 = 𝑓 ↔ (𝑔‘𝐴) = (𝑓‘𝐴)))
62	52, 61	mpbird 247	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 = 𝑓)
63		simp2 1062	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓 ∈ 𝑋)
64	62, 63	eqeltrd 2701	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 ∈ 𝑋)
65	64	3exp 1264	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 Fn 𝐶) → (𝑓 ∈ 𝑋 → ((𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋)))
66	7, 33, 65	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → (𝑓 ∈ 𝑋 → ((𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋)))
67	66	rexlimdv 3030	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → (∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋))
68	31, 67	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → 𝑔 ∈ 𝑋)
69	68	ralrimiva 2966	. . 3 ⊢ (𝜑 → ∀𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)𝑔 ∈ 𝑋)
70		dfss3 3592	. . 3 ⊢ ((ran ∪ 𝑋 ↑𝑚 𝐶) ⊆ 𝑋 ↔ ∀𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)𝑔 ∈ 𝑋)
71	69, 70	sylibr 224	. 2 ⊢ (𝜑 → (ran ∪ 𝑋 ↑𝑚 𝐶) ⊆ 𝑋)
72	6, 71	eqssd 3620	1 ⊢ (𝜑 → 𝑋 = (ran ∪ 𝑋 ↑𝑚 𝐶))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  {csn 4177  ∪ cuni 4436  ∪ ciun 4520  ran crn 5115   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by: (None)